I really don't like the term "inherit" here, it leads to a lot of confusion.
Java only allows interfaces to extend other interfaces, and for classes to implement interfaces.
If you look at an interface as a mathematical set of declarations, then each "extends" merely provides the union of the set from the superinterface and that of the current interface. You are therefore allowed to do multiple "unions".
When you eventually get to a class that implements one or more interfaces, the semantics here are merely that the class must provide implementations for all the methods in the set. A class implementing multiple interfaces could be rewritten as a class implementing a single interface that extends all the above interfaces.
In the case of classes inheriting multiple classes it is not allowed because it leads to a variety of problems, including the diamond problem. For instance, if I have two supertypes with different implementations of the same method signature, which one should be used in the subtype?